where the first summand is the kinetic action and the second the gauge interaction term.

The above action functional is called the Nambu-Goto action in dimension 1. Alternatively (and mandatorily for vanishing mass parameter), the kinetic action is replaced by the corresponding Polyakov action.

Properties

Covariant phase space

We assume for simplicity that the class of the background circle bundle is trivial, so that the connection is equivalently given by a 1-form A∈Ω1(X)A \in \Omega^1(X). Write F=dAF = d A for its curvature 2-form: the field strength of the electromagnetic field.

Proposition

Proof

Let ℝd→≃U↪X\mathbb{R}^d \stackrel{\simeq}{\to} U \hookrightarrow X be a local coordinate patch with coordinates {xμ}\{x^\mu\} and assume that γ\gamma takes values in UU (or at least that its variation is supported there, which we can assume without restriction of generality). Then the variation is given by is

To deal with this, we first look at variations of trajectories in a small region where g(γ˙,γ˙)g(\dot \gamma, \dot \gamma) is non-zero. For such we can always find a diffeomorphismΣ→≃Σ\Sigma \stackrel{\simeq }{\to} \Sigma such that this term is constantly =1= 1 in this region (recall that configurations are diffeomorphism classes of smooth curves, so we may apply such a diffeomorphism at will to compute the variation).

Proposition

With the above choice of diffeomorphism gauge, the equations of motion are